FREQUENCY STANDARDS, CHARACTERIZATION 767

linear Frequency Drift

When linear frequency drift exists (i.e., y ( t ) = d t ) , no tractable model exists for the power spectral density S,(f). Direct calcu- lation in the Lime domain using Eqs. (17) and (34) yields

d a,(r) = - T (42) J z

and

d Mod u,(T) = -T J z (43)

respectively. Thus linear frequency drift yields T + ~ law for both q ( d and Mod q&d.

OTHER MEASURES OF FREQUENCY STABILITY

A number of other mt.asures have been proposed and used during the past 25 years. Each measure has some advantages and limitations compared with the well-established power spectral density and the Allan variance. The Hadamard vari- ance (11) has been developed for high-resolution spectral analysis of y ( t ) from measurements of yk. The high-pass vari- ance (12) has been proposed through the transfer function ap- proach and is defined by

It was shown that the Allan variance can be estimated by high-pass filtering the demodulated phase noise without us- ing counting technique. A band-pass variance (12) has also been proposed to distinguish white and flicker phase noise processes. A filtered Allan variance (13) has been used to sep- arate various noise processes.

BIBLIOGRAPHY

1. Proceedings of the IEEE-NASA Symposium on Short Term Fre- quency Stability, Goddard Space Flight Center, Greenbelt, MD, Nov. 23-24, 1964; Document NASA SP-80.

2. Special issue on frequency stability, Proc. IEEE, 54: 101-338, 1966.

3. J. A. Barnes et al., Characterization of frequency stability, NBS Technical Note 394, US Government Printing Office, Washington, D.C., October 1970; also IEEE Trans. Instrum. Meas., IM-20:

4. P. Kartaschoff, Frequency and Time, London: Academic Press, 1978.

5. J. Rutman and F. L. Walls, Characterization of frequency stabil- ity in precision frequency sources, Proc. IEEE, 79: 952-960, 1991.

6. G. Becker, m e r die Begriffe Phase-Phasenzeit und Phasenwin- kel bei zeitabhangigen Vorgiingen, PTB-Mitteilungen, 81: 348- 352, 1971.

7. D. W. Allan, Statistics of atomic frequency standards, Proc. IEEE,

8. J. J. Snyder, Algorithm for fast digital analysis of interference fringes, Appl. Opt., 19: 1223-1225, 1980.

9. D. W. Allan and J. A. Barnes, A modified “Allan variance” with increased oscillator characterization ability, In Proceedings of the 35th Annual Frequency Control Symposium, 1981, pp. 470-475.

105-120, 1971.

54: 221-230, 1966.

10. P. Lesage and T. Ayi, Characterization of frequency stability: analysis of the modified Allan variance and properties of its esti- mate, IEEE Trans. Instrum. Meas., IM-33: 222-225, 1971.

11. R. A. Baugh, Frequency modulation analysis with the Hadamarc! variance. In Proceedings of the 25th Annual Frequency Control Symposium, 470-475, 1981.

12. J. Rutman, Characterization of frequency stability: A transfer function approach and its application to measurement via filter- ing of phase noise, IEEE Trans. Instrum. Meas., IM-23: 40-48, 1974.

13. J. Groslambert et al., A new “filtered Allan variance” and its ap- plication to the identification of phase and frequency noise sources, In Proceedings of the 43rdAnnual Frequency ControZSym- posium, 326-330, 1989.

HIDEMI TSUCHIDA Electrotechnical Laboratory

FREQUENCY STANDARDS, CHARACTERIZATION

In this article we describe the characterization of frequency standards following the general definitions accepted by the IEEE, the International Telecommunication Union-Radio- communications (ITU-R), and International Radio Consulta- tive Committee (CCIR) (1-3). In using the term “frequency standard” we imply that changes in the frequency A v of the device are small compared to its nominal frequency vo and that therefore the frequency would be about the same if we were to remeasure it. This permits us to treat the variations in fractional frequency Avlv , as small compared to 1, and this greatly simplifies the mathematics of the characterization. We describe what is meant by accuracy and frequency stabil- ity of a frequency standard. The variations in frequency (which define the frequency stability) can be classified into two basic types, random and systematic. To characterize the random variations in frequency the systematic effects must be removed from the data. Special statistical techniques other than the standard variance must then be used to quantify the random variations in frequency, because some of the noise processes are not stationary. By this we mean that the mean of the frequency noise changes slowly with time. As part of the statistical treatment we describe how to determine the confidence intervals for the estimates of the various types of random frequency noise.

ACCURACY

Frequency standards generate a periodic voltage signal, whose ideal frequency is defined as a specific number of oscil- lations per second. The second is the agreed unit of time, based on the energy difference between two energy levels of the unperturbed cesium atom (41, and it is this definition that allows the specification of accuracy of a frequency standard. Generally there will be offsets or biases in the actual fre- quency of a standard when compared to the ideal or defined value (according to the definition of the second) due to system- atic and random effects (5-7). The accuracy of a frequency standard, more recently described as an uncertainty, is a

Work of US. Government. Not subject to copyright.

768 FREQUENCY STANDARDS, CHARACTERIZATION

measure of the confidence relative to a physical model of the standard and the evaluation process. To evaluate the accu- racy of a frequency standard, a list of known sources of fre- quency offsets is made, and the offset or bias due to each source and its unceitainty are carefully measured or com- puted. The uncertainty is a proper summation of the esti- mates of various systematic offsets and random noise (8). For

10 MHz and an uncertainty of has an frequency error of 5 -t 0.1 Hz.

example, a frequency standard with an output frequency of

STAB I LlTY

Figure 1 shows the output voltage signal of an ideal frequency standard as a function of time. The maximum value Vo is the nominal amplitude of the signal. The time required for the signal to repeat itself is the period T of the signal. The nomi- nal frequency vo of the signal is the reciprocal of the period, 1/T. This voltage signal can be represented mathematically by a sine function,

u ( t ) = V, sine = V, sin(2nuot) (1)

where the argument e = 2mgt of the sine function is the nomi- nal phase of the signal. The time derivative of the phase e is 27w0 and is called the nominal angular frequency wo. In the frequency domain, this ideal signal is represented by a delta function located at the frequency of oscillation. Since this sig- nal is ideal, there are no known sources that cause frequency shifts, and thus its frequency is totally accurate and its uncer- tainty is 0. Furthermore, its frequency, phase, and amplitude are constant; therefore the signal is also stable in frequency, phase, and amplitude.

In real situations, the output signal of an oscillator (source) or frequency standard has noise. Such a noisy signal is illustrated in Fig. 2. In this example we have depicted a case where the noise power is much less than the signal power. Frequency instability is the result of fluctuations in the period of the signal. Amplitude instability is the result of fluctuations in the peak values of the voltage. Phase instabil- ity is the result of fluctuations in the zero crossings. Since the period (and thus the frequency) of the signal is related to its phase, frequency instability and phase instability are di- rectly related.

Figure 3 shows the power spectrum of a noisy signal (power as a function of frequency) as measured by a spectrum ana-

+

0, or s o P

-

I

Time

Figure 2. Output voltage of a noisy signal.

lyzer. Although the maximum power occurs at the frequency of oscillation, other peaks are observed at frequencies of 2u,, 3u0, . . ., nvo, where n is a positive integer. These frequencies are called harmonics of the fundamental frequency yo; 2u0 is the second harmonic, 3u0 is the third harmonic, and so on. The power a t these harmonic frequencies will depend on the design of the source. The spectrum around the fundamental frequency displays power sidebands a t frequencies above the carrier (upper sideband) and a t frequencies below the carrier (lower sideband). These power sidebands are the result of phase fluctuations and amplitude fluctuations in the signal. While the power spectrum gives an idea of the total noise of a signal, it does not give information about the relative mag- nitude of its phase instabilities and amplitude instabilities. Furthermore, at frequencies close to uo it is difficult to sepa- rate the noise power from the power of the fundamental fre- quency. Therefore, special measurement techniques are needed to measure phase instabilities and amplitude instabil- ities in sources.

A noisy signal can be mathematically represented by

u ( t ) = W, + c( t ) l sin[2nuot + 4 ( t ) l (2)

where d t ) represents amplitude fluctuations (amplitude devi- ation from the nominal amplitude Vo) and &t) represents phase fluctuations (phase deviation from the nominal phase 2nu,$) (1). The instantaneous frequency of this signal is de- fined as

1 d 1 d u ( t ) = - 2n - dt (phase) = u, + - 2n -$( t ) dt (3)

Time

Figure 1. Ideal output voltage of a frequency standard.

Frequency Figure 3. Power spectrum of a noisy signal.

Frequency fluctuations refer to the deviation d t ) - vo of the instantaneous frequency from the nominal frequency. Frac- tional frequency fluctuations, denoted as y( t ) , refer to fre- quency fluctuations normalized to v,,; that is,

(4)

Equation (4) indicates that there is a direct relation between phase fluctuations and fractional frequency fluctuations. Therefore, if a signal exhibits a certain amount of phase fluc- tuation, it also exhibits frequency fluctuation given by Eq. (4). The time deviation x( t ) of a signal is the integral of y ( t ) from 0 to t . Thus one can write

The frequency stability of a frequency standard is affected by random noise processes and by systematic, deterministic changes. Systematic effects often dominate the frequency sta- bility in nonlaboratory environments. The sensitivity of the standard to temperature, humidity, atmospheric pressure, magnetic field, and radiation may play a role (9-11). Gener- ally, frequency (and thus phase) stability is divided into three regions: short-term, medium-term, and long-term stability. (Though the term frequency stability is used throughout most of the literature, the term actually refers to instabilities in the frequency.)

Short-term frequency stability refers to the random, nonsys- tematic fluctuations that are related to the signal-to-noise ra- tio of the device. In quartz crystal resonators this refers to the region dominated by white phase noise, where the time between observed frequencies (sample time) is less than a sec- ond. In atomic frequency standards the short-term stability also includes white frequency noise and extends to sampling times of several minutes.

Medium-term frequency stability refers to the region where flicker noise dominates. The sampling time characteristic of this region is a function of the type of frequency standard. Short-term and medium-term frequency stability can be char- acterized either in the frequency domain or in the time do- main, after known systematic effects have been removed. Fre- quency-domain characterization and measurements are generally used when the sample time of interest is less than a second. For sampling times longer than a second, time-do- main measurements are used to characterize frequency sta- bility.

Long-term frequency stability includes random-walk fre- quency-noise processes in addition to systematic, determinis- tic changes in frequency observed when the sampling time is long. The long-term, systematic frequency change is called frequency drift (2,12). Drift includes frequency changes due to changes in the components of the source in addition to those due to external parameters such as temperature, humidity, pressure, magnetic field, and radiation (9- 11). Frequency aging, on the other hand, refers to the long-term systematic frequency change due to changes in the components of the source, independent of parameters external to the source (2,121.

FREQUENCY STANDARDS, CHARACTERIZATION 769

Frequency-Domain Characterization

Phase fluctuations in the frequency domain, or phase modula- tion (PM) noise, are characterized by the power spectral den- sity (PSD) of the phase fluctuations, given by (13)

where 9[ ] is the Fourier transformation and &',+(T) is the autocorrelation function of the phase fluctuations given by

A more practical definition of PSD[&(t)l is

where [4(f)lZ is the mean squared phase deviation at an offset frequency f from the frequency vo (called the carrier in this context), and BW is the bandwidth of the measurement sys- tem (1,13-15). The offset frequency f is also called the Fou- rier frequency. The units for S , ( f ) are rad2/Hz. Equation (8) is defined for 0 < f < m ; nevertheless it includes fluctuations from the upper and lower sidebands and thus is a double- sideband measure.

The PM noise measure recommended by the IEEE (1,14,15) is .L ( f ) , defined as

At Fourier frequencies far from the carrier frequency, where the integrated PM noise from 03 to f (the Fourier frequency) is less than 0.1 rad2, d ( f ) is equal to the single-sideband phase noise. The units for 1 ( f ) are decibels below the carrier in a 1 Hz bandwidth (dBc/Hz).

Frequency fluctuations in the frequency domain, or fre- quency modulation (FM) noise, are characterized by the power spectral density of the fractional frequency fluctuations, given by

where y( f)' represents the mean squared fractional frequency deviation at an offset (Fourier) frequency f from the carrier (1,13-15). S,( f ) is defined for Fourier frequencies 0 < f < m.

and its units are inverse hertz. The conversion between S,(f) and S + ( f ) can be obtained

from Eq. (4). Applying the Fourier transformation to both sides of Eq. (4), squaring, and dividing by the measurement bandwidth results in

Amplitude fluctuations in the frequency domain, or ampli- tude modulation (AM) noise, are characterized by the powei

i 7 0 FREQUENCY STANDARDS, CHARACTERIZATION

White phase (fa) I I I I I I

log 0

Random-walk

Flicker frequency (f-')

Flicker phase (f ')

White frequency (f O)

log f Figure 4. PM and FM noise characteristics of a source.

spectral density of the fractional amplitude fluctuations, given by

S , ( f ) =PSD - = - (12) [ ( & where 4 f ) ' represents the mean squared amplitude deviation at an offset frequency f from the carrier (1). S , ( f ) is defined for Fourier frequencies 0 < f < Q), and its units are inverse hertz.

In free-running sources, the FM noise is usually modeled by the sum of five different power laws or noise types as

where h-2f-2 represents random-walk frequency noise, h- , f - ' represents flicker frequency noise, hop represents white fre- quency noise, h l f l represents flicker phase noise, and h 2 f rep-

resents white phase noise. Similarly, the PM noise can be modeled by

S , ( f ) = C k p f @ = k - J 4 + k - 3 f - 3 0

p=-4

+ K - 2 f - 2 + k- , f - ' + k o P (14)

where k-*f-* represents the random-walk frequency noise, k-3 f - 3 represents flicker frequency noise, k - z f - 2 represents white frequency noise, k - l f - ' represents flicker phase noise, and kofo represents white phase noise. Notice that SJf l and S,(fl have different slopes for a specific type of noise, as im- plied by Eq. (11). Equation (11) can be used to obtain the con- version between the S J f ) and S J f ) coefficients, yielding

k , =$ha for p = a - 2 (15)

Figure 4 shows the common noise types characteristic of the PM noise and the FM noise of a source (1,14-17). Usually a source exhibits two or three of the noise types shown in the plots (17).

The AM noise of a source can typically be modeled by the sum of three different power laws or noise types:

0

SaCf) = h a y = h-2f -2 + h - , f - ' + hofo (16) a = - 2

where h-zf-2 represents random-walk amplitude noise, h-,f-' represents flicker noise, and hop represents white amplitude noise (18). Figure 5 shows the common noise types character- istic of the AM noise of a source.

Upper and lower PM sidebands are always equal and 100% correlated. Likewise the upper and lower AM sidebands are always equal and 100% correlated. This is true even when the RF spectrum is not symmetric about the carrier (18a). The phase between AM and PM noise varies randomly with time for broadband additive noise (18a).

Time-Domain Characterization

In the time domain, the fractional frequency stability of a sig- nal is usually characterized by the Allan variance, a type of two-sample frequency variance given by (1,13-15)

c

log f Figure 5. AM noise characteristics of a source.

FREQUENCY STANDARDS, CHARACTERIZATION 771

where 7, is the average fractional frequency of interval by

given

In practical situations only a finite number of fractional fre- quency samples are available, and the Allan variance is ap- proximated by

(19)

where M = N - 1 is the number of frequency samples (N is the number of time samples) (1,14,15). The Allan variance can also be expressed in terms of time samples using Ft =

(X,+1 - X J %

The square root of the Allan variance is called the Allan devi- ation, a,(~).

When time samples are taken every T,, seconds, the Allan variance can be computed for several sampling times T = n~~ where n > 0. For n > 1, overlapped samples can be used to compute 4 as shown in Fig. 6, providing better confidence intervals (1,19). A n expression for the fully overlapped Allan variance can be derived using Fig. 6 and Eq. (19). For T =

n~~ Eq. (19) becomes

which can also be expressed in terms of time-domain data by substituting 7, for (x,+" - x , ) / T z T ~ :

Figure 7(a) shows a log-log plot of the Allan variance as a function of the sampling time T for a source that exhibits all five common noise types. The slopes of the white PM noise and the flicker PM noise are the same; therefore these two noise types cannot be separated using this plot. The Allan variance can often be modeled by the sum of four different power laws:

Figure 6. Computation of G,} for the overlapped Allan variance and n = 2.

White -1

-

-

Random-walk

\ phase (F2) I -

-

Random-walk

frequency (r O) I I I I I I

log

\ White phase ( r-3) -1 -

Random-wal k

frequency (r O ) I I I I I I

log ( 5 )

(b)

Figure 7. a',(~) and Mod 4 ( ~ ) for the five noise types

where p-2r-2 represents white phase and flicker phase noise, p _ , ~ - l represents white frequency noise, pO?D represents flicker frequency noise, and p171 represents random walk frequency noise.

When the dominant noise type in the short term is flicker PM or white PM, the modified Allan variance can be used to improve the estimate of the underlying frequency stability of the sources (1,14,20). Here a new series is created by aver- aging n adjacent phase (time) measurements of duration T ~ . The average fractional frequencies are computed from the {TL}, as illustrated in Fig. 8. For N time samples and T = n ~ ~ , the resulting modified Allan variance is

Equation (24) can also be expressed in terms of the initial time-domain data { x ~ } :

Mod at ( r )

(25)

772 FREQUENCY STANDARDS, CHARACTERIZATION

Figure 8. Computation of G,} and G,} for the Modified Allan variance and n = 3.

where

and

(26)

(27)

Here is the phase (time) averaged over n adjacent measure- ments of duration T ~ . Thus Mod uY(d is proportional to the second difference of the phase averaged over a time nTo.

Viewed from the frequency domain, Mod uY(d is proportional to the first difference of the frequency averaged over n adja- cent samples. The square root of the modified Allan variance is called the modified Allan deviation Mod ~ " ( d .

Figure 9 shows the ratio R(n) = [Mod ~ ( T ) ] / < ( T ) as a func- tion of n for all five types of noise processes (15). For random- walk FM, flicker FM, and white FM the ratio is constant for n 5 5 . Therefore, Mod o$(T) and 4 ( ~ ) have the same slope for these noise types. For white PM noise (a = 2), the slope is - l /n; therefore the slope of Mod $(TI is equal to the slope of $(T) divided by T. Finally, for flicker PM noise the ratio

asymptotically reaches 3.37/(1.04 + 3 In W ~ T . In this case and for n > 10, the slope of Mod $(T) is approximately 7-l. Figure 9 also shows that Mod 4(d is considerably smaller than @T) far white PM and flicker PM noise. Not only does Mod &T) provide a different slope for white PM noise and flicker PM noise, allowing the separation of the two noise processes (see Fig. 7b); it can also speed the stability measurements. If a system is limited by white and flicker PM noise a t short average times, using Mod a",(~) reduces the measurement time required to observe white FM, flicker FM, and random-walk FM a t longer averaging times, in comparison with that re- quired when using O$T) (15).

At long averaging times when the ratio N T & h T , ) is close to 1, the Allan variance has a bias related to its insensitivity to odd noise processes in the phase (time) fluctuations (odd with respect to the midpoint). In these situations an extension of the Allan variance that removes this bias can be used to characterize the frequency stability of a source. This vari- ance, $,TOTa(~), is obtained by extending the {x,} in both direc- tions and then computing the Allan variance from the new {x:} sequence (21-23). Figure 10 illustrates this extension of {xL}: on the left side the extension is the inverted mirror image of {q} with respect to x l ; on the right side it is the inverted mirror image of {x,} with respect to xN. How far this extension depends on the maximum value n, of n. For N time data points, n, is the integer part of (N - 1)/2. The far-left data point is xi -nm = 2x1 - xnm; the far-right data point is x ; + , ~ - ~ = 2xN - xN-nm+l. Thus ~ , T O T ~ ( T ) is given by

Equation (28) can be expressed in terms of {x:} using T,' = (xI'+" - x{ ' ) /T as

For a detailed description of 0 2 , , T o T a ( ~ ) see Refs. 21-23.

I 1 I I 1 1 " -

- - -

- -

5 - - - -

2 - -

Figure 9. Ratio of the modified Allan o,ol I I I I I I I I I I I I I l l 1 1

variance to the Allan variance, R(n) = 1 2 5 10 2 5 100 2 5 1000 [Mod ~ ( T ) I / ~ ( T ) , as a function of n (15). T = ni-,. Number of samples averaged, n

- 2 - 1 0 1 2 3 4 5 6 7 8 9 1 0

Figure 10. Extension of {xt} for u:(T).

CONVERSION BETWEEN TIME-DOMAIN MEASURES AND FREQUENCY-DOMAIN MEASURES

Frequency-domain data S,( f , can be converted to time-do- main data 0 3 7 ) using the relation (1,241

(30)

Equation (30) is derived by expressing both S,(f) and O$(T) in terms of the autocorrelation function ??,(T) of the random process &(t) and then combining the two expressions to cancel .8 ,(TI (13). Similarly,

Expressions for Mod 037) (15,241, obtained using a similar procedure, are

The inclusion of f h as the upper limit of the integral as- sumes that the term inside the integral is multiplied or “fil- t e r ed by an infinitely sharp low-pass filter with cutoff fre- quency f h . Table 1 shows the results of Eqs. (31) and (33) for the five types of noise for 2 n f h 7 * 1 (1,14-16,241. The results will depend on the type of filter assumed. While an infinitely sharp filter was assumed in Eqs. (3014331, actual measure- ment systems have different filter response.

Table 1. Conversion Factors for a:(f) and Mod cr : ( f ) (1, 14-16. 24)

Noise Type S , ( f ) u;(f) Mod u:(f) Random walk h-2f-Z (2n2~/3 )h-2 5 .42h .*~

Flicker frequency h . f ’ (2 In 2)h-, 0.936h-, White frequency hofo hJ2 7 h0/47

frequency

1.038 + 3ln(%7) 1 3.37 1 Flicker phase h i f l 4n2 h ’ 2 g h 1 - ; i

White phase

FREQUENCY STANDARDS, CHARACTERIZATION 773

Table 2. Empirical Equations for the Number of Degrees of Freedom When Computing Confidence Intervals for the Overlapped Allan Variance (16)

~ ~~

Noise Type Q No. of Degrees of Freedom

White phase

Flicker phase

(N + 1)(N - 2n) 2(N - n ) -2

Flicker frequency +1

N - 2 ( N - 1)2-3n(N- 1 ) + 4 n 2 +2 - Random-walk frequency n (N - 3)’

Expressions have also been derived for a single-pole filter (15,25,26). These expressions, along with those in Table 1. constitute the boundaries for (T:(T) and Mod ( T ~ ( T ) , given a spe- cific PSD of phase fluctuations (15). For this reason it is im- portant to specify the filter frequency response, including the high cutoff frequency, when specifying the Allan and modified Allan variances of a source.

Generally, conversion from (T:(T) or Mod ( T ~ ( T ) to the fre- quency domain is not possible, unless specific information about the noise characteristics is known. Greenhall demon- strated that several different spectral densities of random processes can have the same Allan variance (27). However, in the case where the spectral density follows the noise model in Eqs. (13)-(14), a one-to-one correspondence between S,(f) and 4 7 ) and Mod o ~ ( T ) is found, except that for white PM and flicker PM noise ( T ~ ( T ) exhibits the same slope, corre- sponding to T+. Often, uniqueness fails more generally. Some sources have internal phase-locked loops, and their noise spectra deviates from the model in Eqs. (13)-(14) (28); others exhibit 60 Hz and other peaks that will affect ( T ~ ( T ) (16,29). Generally, multivariance analysis should be used to obtain frequency-domain coefficients for each type of noise from time-domain data (30).

CONFIDENCE INTERVALS FOR ~~$71 AND Mod 4(7)

The Allan variance is defined as the first difference of average fractional frequencies, averaged over an infinite time. Since only a finite number M of frequency samples can be taken, we can only estimate the Allan variance and deviation, and the confidence of this estimate depends on the number of samples.

A simple method to obtain confidence intervals is to use the chi-squared distribution function. “he Allan variance has a chi-squared distribution function given by

(34)

where df is the number of degrees of freedom (16). The Allan variance is the sum of the squares of the first differences of adjacent fractional frequency values. If all the first-difference values were independent, then the number of degrees of free- dom would be equal to the number of first difference values. This is not the case, and thus other procedures have been used to compute the number of degrees of freedom for TI (16). Table 2 shows analytical (empirical) equations that ap-

774 FREQUENCY STANDARDS, CHARACTERIZATION

Table 3. Confidence Intervals for the Nonoverlapped and Fully Overlapped q , ( ~ ) (16) and for the Fully Overlapped Mod u,(T) (31-33)”

Confidence Interval (%)

n

Fullyoverlapped Nonoverlapped u,( T) Fullyoverlapped cy( T) Mod u,(d

+ - + - + Noise Type -

2 White PM 4.1 4.8 2.9 3.2 3.1 3.4 8 7.7 10.1 2.9 3.2 5.2 6.1

32 13.6 23.1 3.0 3.4 9.7 14

2 Flicker PM 3.7 4.3 2.9 3.1 3.0 3.3 8 7.1 9.0 3.6 4.0 5.7 6.8

32 12.7 20.7 5.2 6.1 11 16

2 White FM 3.6 4.0 2.8 3.0 3.0 3.2 8 6.8 8.6 4.8 5.6 5.8 7.0

32 12.5 20.1 8.8 12 11 16

2 Flicker FM 3.2 3.5 2.6 3.0 2.9 3.2 8 6.1 7.4 5.1 6.0 5.8 7.1

32 11.1 16.8 9.9 14 11 16

2 Random-walk FM 3.0 3.3 3.0 3.3 3.2 3.5 8 5.7 6.8 5.7 7.0 6.4 8.0

32 10.4 15.2 11 16 12 19

a Confidence intervals for the nonoverlapped q.(d were obtained using df in Table 2 (16). The degrees of freedom used for the fully overlapped d r ) were computed using numerical methods and are approximately equal to those obtained using Table 2 (16, 19). Confidence intervals for the fully overlapped Mod r,(d were obtained from Ref. 33. N = 1025.

proximate the number of degrees of freedom for the fully over- lapped Allan variance (16). The equation depends on the noise type. For nonoverlapped estimates, n in Table 2 is equal to 1, and N refers to the equivalent number of time samples for 7 = n~~ given by Int((N - l ) / n ) + 1, where Int( ) refers to the integer part.

Usually a (p x 100) % confidence interval is computed, where p is the probability that the true Allan variance or Al- lan deviation is within the computed confidence interval. The ( p X 100) % confidence interval for the overlapped Allan vari- ance is given by

where the chi-squared value ,$(C) for (p x 100) % confidence can be obtained from chi-squared distribution tables or from several computer programs. The ( p x 100) % confidence inter- val for the Allan deviation is

The chi-squared distribution can also be used to find the con- fidence intervals for Mod 4(7).

Walter (31) and Greenhall (32) have derived expressions for the number of degrees of freedom of Mod 4(7) using differ- ent procedures. These expressions are complicated and will not be presented here. The two methods yield similar results (33). Table 3 shows the confidence intervals for CT,,(T) with no

overlap and full overlap (16.191, and for Mod q,(d (31-33) for the five noise types. In general, the confidence intervals for the fully overlapped ~ ~ ( 7 ) are smaller than those for the non- overlapped CT,~T) . For random-walk FM noise, the confidence intervals for the nonoverlapped and the fully overlapped q,(d are approximately the same, although Table 3 shows a small degradation when using fully overlapped estimates. This degradation is due to the approximations used in the analytical expressions. Table 3 also shows that the confidence intervals for the fully overlapped Allan deviation are smaller than the ones for the fully overlapped modified Allan devia- tion. Nevertheless, the modified deviation is generally smaller than the Allan deviation, and thus the absolute confidence intervals for the two are similar.

BIBLIOGRAPHY

1. E. S. Ferre-Pika1 et al., Draft revision of IEEE Std 1139-1988: Standard definitions of physical quantities for fundamental fre- quency and time metrology-random instabilities, Proc. 1997 ZEEE Znt. Freq. Control Symp.. 1997, pp. 338-357.

2. R. L. Sydnor and D. W. Allan (eds.), Handbook Selection and Use ofPrecise Frequency and Time Systems, International Telecommu- nication Union, Geneva, Switzerland, 1997.

3. International Radio Consultative Committee (CCIR), Report 580, Characterization of Frequency and Phase Noise, 1986, pp.

4. H. Hellwig, Frequency Standards and C1ocks:A Tutorial Zntroduc- tion, Technical Note 616, Washington, DC: U S . National Bureau of Standards, 1977.

5. J. R. Vig and F. L. Walls, Fundamental limits on the frequency stabilities of crystal oscillators, ZEEE Trans. Ultrason. Ferroelectr. Freq. Control, 42: 576-589, 1995.

142-150.

FREQUENCY SYNTHESIZERS 775

6. IEEE Guide for Measurements of Environmental Sensitivities of Standard Frequency Generators, IEEE Std. 1193, Piscataway, NJ: IEEE Press, 1994.

7. 'Vi. D. Lee et al., The accuracy evaluation of NIST-7, IEEE Trans. Instrum. Meas., 44: 120-124, 1995. An example of evaluating the uncertainties of an atomic frequency standard.

8. B. N. Taylor and C. E. Kuyatt, Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results, Techni- cal Note 1297, Washington, DC: National Institute of Standards and Technology, 1994.

9. F. L. Walls, The influence of pressure and humidity on the me- dium and long-term frequency stability of quartz oscillators, Proc. 42nd Annu. Symp. Freq. Control, Baltimore, MD: 1988, IEEE Cat- alog No. 88CH2588-2, pp. 279-283.

In . J. J. Gagnepain, Sensitivity of quartz oscillators to the environ- ment: Characterization methods and pitfalls, IEEE Trans. Ul- trason. Ferroelectr. Freq. Control, 37: 354, 1990.

11. F. L. Walls and J . J . Gagnepain, Environmental sensitivities of quartz oscillators, IEEE Trans. Ultrason. Ferrelectr. Freq. Control,

12. J . R. Vig and T.R. Meeker, The aging of bulk acoustic wave reso- nators, filters and oscillators, Proc. 45th Annu. Symp. Freq. Con- trol, Los Angeles, CA: 1991, IEEE, Catalog No. 91CH2965-2, pp.

13. J. A. Barnes et al., Characterization of frequency stability, IEEE Trans. Instrum. Meas., IM-20: 105-120, 1971.

14. D. W. Allan et al., Standard terminology for fundamental fre- quency and time metrology, Proc. 42nd Annu. Symp. on Freq. Control, Baltimore, MD: 1988, IEEE Catalog No. 88CH2588-2,

15. D.B. Sullivan et al., Characterization of Clocks and Oscillators, Technical Note 1337, National Institute of Standards and Tech- nology, 1990.

16. D. A. Howe, D.W. Allan, and J. A. Barnes, Properties of signal sources and measurement methods, Proc. 35th Annu. Svmp. Freq. Control, Electronic Industries Association 1981, Washington, DC, Philadelphia, PA: 1981, pp. Al-A47; also in Ref. 4.

17. T. E. Parker, Characteristics and sources of phase noise in stable oscillators, Proc. 41st Annu. Symp. Freq. Control, Philadelphia, PA: 1987, IEEE Catalog No. 87CH2427-3, pp. 99-110.

18. L. M. Nelson, C . W. Nelson, and F. L. Walls, Relationship of AM nose to PM noise in selected RF oscillators, IEEE Trans. Ul- trason. Ferroelectr. Freq. Control, 41: 680-684, 1994.

18a. F. L. Walls, Correlation between upper and lower noise side- bands, to be published in Proc. 1998 IEEE Int. Freq. Control Symp., Pasadena, CA, May 1998.

19. S. R. Stein, Frequency and time-their measurement and charac- terization, in E. A. Gerber and A. Ballato (eds.), Precision Fre- quency Control, New York: Academic Press, 1985, vol. 2, chap. 12.

20. D. W. Allan and J. A. Barnes, Modified Allan variance with in- creased oscillator characterization ability, Proc. 35th Annu. Symp. Freq. Control, Electronic Industries Assoc. 1981, Philadelphia,

21. D. A. Howe, An extension of the Allan variance with increased confidence at long-term, Proc. 1995 IEEE Int. Freq. Control Symp., San Francisco, CA: 1995, IEEE Catalog No. 95CH35752, pp. 321-329.

22. D. A. Howe, Methods of improving the estimation of long-term frequency variance, Proc. European Frequency and Time Forum, Swiss Foundation for Research in Microtechnology, Neuchatel, Switzerland: 1997, pp. 91-99.

23. D. A. Howe and C. A. Greenhall, Total variance: A progress re- port on a new frequency stability characterization, Proc. 29th Ann. Precise Time Time Interval PTTI Syst. Appl. Meet., Long Beach, CA: 1997, pp. 39-48.

39: 241-249, 1992.

77-101.

pp. 419-425.

PA: 1981, pp. 470-474.

24. F. L. Walls et al., Time-domain frequency stability calculated from the frequency domain: An update, Proc. 4th Eur. Frequency Time Forum, Swiss Foundation for Research in Microtechnology, Neuchatel, Switzerland: 1990, pp. 197-204.

25. P. Lesage and C. Audoin, Characterization x:d measurement of time and frequency stability, Radio Sci., 14 (4): 521-539, 1979.

26. P. Lesage and T. Ayi, Characterization of frequency stability: Analysis of the modified Allan variance and properties of its esti- mate, ZEEE Trans. Instrum. Meas., IM-33: 332-336, 1984.

27. C. A. Greenhall, Does the Allan variance determine the spec- trum?, Proc. 1997 IEEE Inter. Freq. Control Symp., Orlando, FL: 1997, IEEE Catalog No. 97CH36016, pp. 358-365.

28. F. L. Walls and S. R. Stein, Servo techniques in oscillators and measurement systems, Technical Note 692, Washington, DC: U S . National Bureau of Standards, pp. 1-20, 1976.

29. J. Rutman and F. L. Walls, Characterizatim of frequency stabil- ity in precision frequency sources, Proc. ILh'E, 79: 952-960, 1991.

30. F. Vemotte et al., Oscillator noise analysis: multivariance mea- surement, IEEE Trans. Instrum. Meas., IM-42: 342-350, 1993.

31. T. Walter, Characterizing frequency stability: A continuous power-law model with discrete sampling, IEEE Trans. Instrum. Meas., IM-43: 69-79, Feb. 1994.

32. C. A. Greenhall, Estimating the modified Allan variance, Proc. 1995 IEEE Int. Freq. Control Symp., San Francisco, CA 1995, IEEE Catalog No. 95CH35752, pp. 346-353.

33. M. A. Weiss et al., Confidence on the modified Allan variance, Proc. 9th Eur. Frequency Time Forum, Besanqon, France, 1995, pp. 153-165.

EVA S. FERRE-PIKAL FRED L. WALLS National Institute of Standards and

Technology

FREQUENCY SYNTHESIZERS

Frequency synthesis is the engineering discipline dealing with generation of a single or of multiple tones, all derived from a common time-base, always a crystal oscillator.

Traditionally, frequency synthesis (FS) generated a single tone with variable frequency or amplitude. There are new ap- plications, especially for testing and simulation, that require multitone generation a n d even arbi t rary wave generation, which relate to digital frequency synthesis. In the last two decades, FS has evolved from mainly analog to a mix of ana- log, radio-frequency (RF), digital, a n d digital signal pro- cessing (DSP) technologies.

There are three major FS techniques:

1. Phase Lock Loop. Also known as indirect synthesis, the most popular FS technique, based on a feedback mecha- nism that enables simplicity a n d economics via digital division a n d analog processing.

2 . Direct Analog. An analog technique using multiplica- tion, division, a n d mix-filtering, offers excellent signal quality a n d speed.

3. Direct Dig i ta l . DSP method that generates a n d manip- ulates the signal in the numbers (digital) domain and eventually converts to its analog form via a digital to analog converter.